From affine manifolds to complex manifolds: instanton corrections from tropical disks

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1 UCSD Mathematics Department From affine manifolds to complex manifolds: instanton corrections from tropical disks Mark Gross Directory Table of Contents Begin Article Copyright c 2008 mgross@math.ucsd.edu Last Revision Date: May 25, 2008

2 Section 1: Affine manifolds and mirror symmetry 2 1. Affine manifolds and mirror symmetry Definition. An affine manifold B is a real manifold with charts ψ i : U i R n such that ψ i ψ 1 j Aff(R n ) for all i, j. An affine manifold is integral if ψ i ψj 1 Aff(Z n ) for all i, j.

3 Section 1: Affine manifolds and mirror symmetry 3 Given an integral affine manifold B, we can define a family of lattices Λ T B locally generated by / y 1,..., / y n, where y 1,...,y n are local affine coordinates. Similarly we define ˇΛ T B locally generated by dy 1,..., dy n. Definition. X(B) := T B /Λ ˇX(B) := TB/ˇΛ

4 Section 1: Affine manifolds and mirror symmetry 4 X(B) carries a canonical complex structure: If y 1,...,y n are local affine coordinates on B, x 1,...,x n coordinates on T B so that (x 1,..., x n, y 1,..., y n ) corresponds to the tangent vector n x i, y i at the point with coordinates i=1 (y 1,..., y n ), then complex coordinates are given by z i = e 2π 1(x i+ 1y i). These coordinates define a well-defined complex structure, independent of the choice of coordinates.

5 Section 1: Affine manifolds and mirror symmetry 5 ˇX(B) carries a canonical symplectic structure: TB always carries a canonical symplectic form, and this symplectic form descends to a symplectic form on ˇX(B).

6 Section 1: Affine manifolds and mirror symmetry 6 This gives torus fibrations X(B) ˇX(B) B B Toy version of the SYZ conjecture. Mirror symmetry is the correspondence between complex manifolds X(B) and symplectic manifolds ˇX(B).

7 Section 1: Affine manifolds and mirror symmetry 7 Problem. There are very few interesting compact examples of mirror symmetry arising in this manner, e.g. complex tori. To get more interesting examples, we need to consider integral affine manifolds with singularities. This is a manifold B with an open subset B 0 B with an integral affine structure, with := B \ B 0 codimension two in B.

8 Section 1: Affine manifolds and mirror symmetry 8 Example. There exists a three-dimensional integral affine manifold with singularities B such that X(B 0 ) B 0 and ˇX(B 0 ) B 0 can be compactified topologically to torus fibrations X(B) B and ˇX(B) B with X(B) homeomorphic to the quintic three-fold and ˇX(B) homeomorphic to the mirror quintic. [G., Ruan, Haase and Zharkov].

9 Section 1: Affine manifolds and mirror symmetry 9 Generally true. Given reasonable restrictions on the singularities of B, X(B 0 ) and ˇX(B 0 ) can be compactified topologically.

10 Section 1: Affine manifolds and mirror symmetry 10 Generally true. Given reasonable restrictions on the singularities of B, X(B 0 ) and ˇX(B 0 ) can be compactified topologically. Expectation. Given reasonable restrictions on the singularities of B, ˇX(B0 ) can be compactified symplectically. [Symington, dim B = 2, Castano-Bernard, Matessi, dimb = 3.]

11 Section 1: Affine manifolds and mirror symmetry 11 Generally true. Given reasonable restrictions on the singularities of B, X(B 0 ) and ˇX(B 0 ) can be compactified topologically. Expectation. Given reasonable restrictions on the singularities of B, ˇX(B0 ) can be compactified sympletically. [Symington, dim B = 2, Castano-Bernard, Matessi, dimb = 3.] Certainty. X(B 0 ) usually cannot be compactified as a complex manifold.

12 Section 1: Affine manifolds and mirror symmetry 12 Definition. For B an integral affine manifold, ǫ > 0, define X ǫ (B) = T B /ǫλ. Reconstruction problem. [G., Kontsevich- Soibelman] Let B 0 be an integral affine manifold with compactification B. Find a family of compact complex manifolds X ǫ (B) for ǫ < ǫ 0 which are compactifications of small deformations of X ǫ (B 0 ) of size O(e C/ǫ ).

13 Section 1: Affine manifolds and mirror symmetry 13 Approaches. (1) Fukaya, Studied deformations of the complex structure on X ǫ (B 0 ) for dim B = 2 using the Kodaira-Spencer equation, and found heuristic arguments to suggest solutions concentrate along trees made of gradient flow lines on B 0 with leaves at the singularities.

14 Section 1: Affine manifolds and mirror symmetry 14 Approaches. (1) Fukaya, Studied deformations of the complex structure on X ǫ (B 0 ) for dim B = 2 using the Kodaira-Spencer equation, and found heuristic arguments to suggest solutions concentrate along trees made of gradient flow lines on B 0 with leaves at the singularities. (2) Kontsevich and Soibelman, Produce a non-archimedean K3 surface (a rigid analytic space) from an affine S 2 with 24 singular points. The description involves corrections coming from trees of gradient flow lines as in Fukaya s approach.

15 Section 1: Affine manifolds and mirror symmetry 15 Approaches. (1) Fukaya, Studied deformations of the complex structure on X ǫ (B 0 ) for dim B = 2 using the Kodaira-Spencer equation, and found heuristic arguments to suggest solutions concentrate along trees made of gradient flow lines on B 0 with leaves at the singularities. (2) Kontsevich and Soibelman, Produce a non-archimedean K3 surface (a rigid analytic space) from an affine S 2 with 24 singular points. The description involves corrections coming from trees of gradient flow lines as in Fukaya s approach. (3) G. and Siebert, (2001-)2007. Produce a flat family X S = Speck[[t]] from B, with X 0 a degenerate Calabi-Yau manifold. A discrete Legendre transform of B produces a new affine manifold ˇB, and we can replace gradient flow lines with straight lines on ˇB. The flat family is described using tropical trees on ˇB. This simplifies the construction, allowing it to work in any dimension.

16 Section 2: A toy example A toy example.

17 Section 2: A toy example. 17 Let B R n be a lattice polytope, and let P be a decomposition of B into lattice polytopes. Let ϕ : B R be a strictly convex piecewise linear function with respect to P, with integer slopes. B

18 Section 2: A toy example. 18 Let B R n be a lattice polytope, and let P be a decomposition of B into lattice polytopes. Let ϕ : B R be a strictly convex piecewise linear function with respect to P, with integer slopes. P

19 Section 2: A toy example. 19 Let B = {(m, r) R n+1 m B, r ϕ(m)}, and let Σ be the normal fan to B, living in (R n+1 ) : B Σ B

20 Section 2: A toy example. 20 We obtain a toric variety X Σ from this fan. The projection onto the last coordinate (R n+1 ) R induces a morphism f : X Σ A 1 with f 1 (t) = P B, the projective toric variety defined by the polytope B, and f 1 (0) is a union of toric varieties P σ for maximal cells σ P.

21 Section 2: A toy example. 21 e.g. P 1 A 1 A 1

22 Section 2: A toy example. 22 Different approach: Build a k-th order deformation of the central fibre: X k Speck[t]/(t k+1 ) by gluing together thickenings of affine subsets of irreducible components of X 0.

23 Section 2: A toy example. 23 A cell τ 2 P defines a projective toric variety which is a stratum of the singular fibre X 0.

24 Section 2: A toy example. 24 A cell τ 2 P defines a projective toric variety which is a stratum of the singular fibre X 0. A choice of a face τ 1 τ 2 determines an affine open subset of this stratum.

25 Section 2: A toy example. 25 For any pair of cells τ 1 τ 2, we will define a ring, which gives a thickening of this open affine subset.

26 Section 2: A toy example. 26 For any pair of cells τ 1 τ 2, we will define a ring, which gives a thickening of this open affine subset. For σ P a maximal cell, let n σ R n be the slope of ϕ.

27 Section 2: A toy example. 27 For any pair of cells τ 1 τ 2, we will define a ring, which gives a thickening of this open affine subset. For σ P a maximal cell, let n σ R n be the slope of ϕ. Define ϕ τ1 : R n R by ϕ τ1 (m) = max{ n σ, m τ 1 σ}.

28 Section 2: A toy example. 28 For any pair of cells τ 1 τ 2, we will define a ring, which gives a thickening of this open affine subset. For σ P a maximal cell, let n σ R n be the slope of ϕ. Define ϕ τ1 : R n R by ϕ τ1 (m) = max{ n σ, m τ 1 σ}. Define the monoid (semi-group) P τ1 := {(m, r) m Z n, r Z, r ϕ τ1 (m)}.

29 Section 2: A toy example. 29 For any pair of cells τ 1 τ 2, we will define a ring, which gives a thickening of this open affine subset. For σ P a maximal cell, let n σ R n be the slope of ϕ. Define ϕ τ1 : R n R by ϕ τ1 (m) = max{ n σ, m τ 1 σ}. Define the monoid (semi-group) P τ1 := {(m, r) m Z n, r Z, r ϕ τ1 (m)}. Set R τ1 = k[p τ1 ].

30 Section 2: A toy example. 30 τ 1 P τ1 = N 2

31 Section 2: A toy example. 31 τ 1 P τ1 = N Z

32 Section 2: A toy example. 32 For σ P a maximal cell, define ord σ (m, r) = r n σ, m, the height of (m, r) above the plane defined by n σ. This is the order of vanishing of the monomial z (m,r) on the irreducible component defined by σ.

33 Section 2: A toy example. 33 by We next define monomials ideals I >k τ 1,τ 2 k[p τ1 ] I >k τ 1,τ 2 = z (m,r) (m, r) P τ1 and ord σ (m, r) > k for some σ τ 2.

34 Section 2: A toy example. 34 by We next define monomials ideals I >k τ 1,τ 2 k[p τ1 ] I >k τ 1,τ 2 = z (m,r) (m, r) P τ1 and ord σ (m, r) > k for some σ τ 2. Set R k τ 1,τ 2 := k[p τ1 ]/I >k τ 1,τ 2.

35 Section 2: A toy example. 35 by We next define monomials ideals I >k τ 1,τ 2 k[p τ1 ] I >k τ 1,τ 2 = z (m,r) (m, r) P τ1 and ord σ (m, r) > k for some σ τ 2. Set R k τ 1,τ 2 := k[p τ1 ]/I >k τ 1,τ 2. This is a thickening of the affine subset of the projective toric variety P τ2 determined by τ 1.

36 Section 2: A toy example. 36 k = 3 τ 1 τ 2 R k τ 1,τ 2 = k[x,y]/(x k+1 )

37 Section 2: A toy example. 37 k = 3 τ 1 = τ 2 R k τ 1,τ 2 = k[x,y]/(x k+1, y k+1 )

38 Section 2: A toy example. 38 τ 1 = τ 2 k = 3 R k τ 1,τ 2 = k[x,y ±1 ]/(x k+1 )

39 Section 2: A toy example. 39 For a vertex v P, there are natural surjections whenever τ 1 τ 2. We can then construct R k v,τ 2 R k v,τ 1 R k v = lim R k v,τ.

40 Section 2: A toy example. 40 R k v = k[x,y, t]/(t xy, t k+1 )

41 Section 2: A toy example. 41 Next, the schemes SpecRv k can be glued together using common open sets. This produces the desired scheme X k Speck[t]/(t k+1 ).

42 Section 2: A toy example. 42 X Σ k[t]/(t k+1 ) k[t]

43 Section 3: Onwards to affine manifolds Onwards to affine manifolds. We now replace B, P, ϕ with the following data: B is an integral affine manifold. P is a decomposition of B into integral lattice polytopes. ϕ is a multi-valued integral strictly convex piecewise linear function, represented on an open cover {(U i, ϕ i )} with ϕ i strictly convex piecewise linear function with integral slopes on U i, with ϕ i ϕ j affine linear on U i U j.

44 Section 3: Onwards to affine manifolds. 44 Where do the monoids P y live?

45 Section 3: Onwards to affine manifolds. 45 Where do the monoids P y live? On B have an extension 0 Z Λ Λ 0 determined by the extension class c 1 Ext 1 (Λ, Z) = H 1 (B, ˇΛ) represented by the Čech 1-cocycle ( Ui U j, d(ϕ i ϕ j ) )

46 Section 3: Onwards to affine manifolds. 46 For any point y B, y τ 1 τ 2 as before, can choose a representative for ϕ in a neighbourhood of y, yielding a splitting Λ y = Λy Z and then defining as before. P y,τ1 Λ y Z

47 Section 3: Onwards to affine manifolds. 47 For any point y B, y τ 1 τ 2 as before, can choose a representative for ϕ in a neighbourhood of y, yielding a splitting Λ y = Λy Z and then defining as before. P y,τ1 Λ y Z Different choices of representatives yield isomorphic choices of P y,τ1.

48 Section 3: Onwards to affine manifolds. 48 For any point y B, y τ 1 τ 2 as before, can choose a representative for ϕ in a neighbourhood of y, yielding a splitting Λ y = Λy Z and then defining as before. P y,τ1 Λ y Z Different choices of representatives yield isomorphic choices of P y,τ1. We then similarly obtain I >k y,τ 1,τ 2 k[p y,τ1 ] R k y,τ 1,τ 2 = k[p y,τ1 ]/I >k y,τ 1,τ 2.

49 Section 3: Onwards to affine manifolds. 49 Parallel transport of monomials: We can use parallel transport in the sheaf to compare monomials defined at different points.

50 Section 3: Onwards to affine manifolds. 50 Parallel transport of monomials: We can use parallel transport in the sheaf to compare monomials defined at different points. In particular, this may depend on a choice of path connecting these two points!

51 Section 3: Onwards to affine manifolds. 51 Parallel transport of monomials: We can use parallel transport in the sheaf to compare monomials defined at different points. In particular, this may depend on a choice of path connecting these two points! However, if B has no singularities, Λ is has no monodromy in a neighbourhood of any given cell.

52 Section 3: Onwards to affine manifolds. 52 Basic fact. If we parallel transport a monomial (m, r) along a straight line in the direction m, the order of (m, r) on maximal cells σ traversed by this line increases! m

53 Section 3: Onwards to affine manifolds. 53 The same construction given for polyhedral B now works for arbitrary integral affine B, and so we get kth order deformations X k Speck[t]/(t k+1 ).

54 Section 3: Onwards to affine manifolds. 54 The same construction given for polyhedral B now works for arbitrary integral affine B, and so we get kth order deformations X k Speck[t]/(t k+1 ). Example. If we take B = R n /Γ for Γ Z n an integral lattice, this construction yields Mumford s degenerations of abelian varieties, (see also [Alexeev], [Alexeev,Nakamura]).

55 Section 4: Introducing singularities Introducing singularities.

56 Section 4: Introducing singularities. 56 Now let B be an integral affine manifold with singularities, still with a polyhedral decomposition (slightly subtle).

57 Section 4: Introducing singularities. 57 Now let B be an integral affine manifold with singularities, still with a polyhedral decomposition (slightly subtle). In 2-dim. case, assume monodromy of singularities is ( )

58 Section 4: Introducing singularities. 58 Now let B be an integral affine manifold with singularities, still with a polyhedral decomposition (slightly subtle). We insist there is an edge through each singular point in the invariant direction.

59 Section 4: Introducing singularities. 59 Problem. Gluing is not well-defined now! z (a,b,r) z (a,b,r) z (a+b,b,r) z (a,b,r) z (a,b,r)

60 Section 4: Introducing singularities. 60 Solution. Modify gluing! z (a,b,r) (1 + z (1,0,0) ) b z (a,b,r) (1 + z ( 1,0,0) ) b z (a,b,r) z (a,b,r)

61 Section 4: Introducing singularities. 61 Solution. Modify gluing! z (a+b,b,r) (1 + z (1,0,0) ) b z (a,b,r) (1 + z (1,0,0) ) b z (a,b,r) (1 + z ( 1,0,0) ) b z (a,b,r) z (a,b,r)

62 Section 4: Introducing singularities. 62 Solution. Modify gluing! z (a+b,b,r) (1 + z (1,0,0) ) b z (a,b,r) (1 + z (1,0,0) ) b z (a,b,r) (1 + z ( 1,0,0) ) b z (a,b,r) z (a,b,r)

63 Section 4: Introducing singularities. 63 How does this help us?

64 Section 4: Introducing singularities. 64 How does this help us? Let s examine the gluing closely, taking ϕ to be given by { 0 if b 0 ϕ(a, b) = b if b 0 and set x = z (0,1,1) y = z (0, 1,0) w = z ( 1,0,0)

65 Section 4: Introducing singularities. 65 The glued ring is the fibred product k[x, y, w ±1 ]/(y k ) (k[x,y,w ±1 ]/(x k,y k )) 1+w k[x, y, w ±1 ]/(x k ) with the two maps given by x, y, w x, y, w and x x/(1 + w) y y(1 + w) w w

66 Section 4: Introducing singularities. 66 The glued ring is the fibred product k[x, y, w ±1 ]/(y k ) (k[x,y,w ±1 ]/(x k,y k )) 1+w k[x, y, w ±1 ]/(x k ) with the two maps given by x, y, w x, y, w and x x/(1 + w) y y(1 + w) w w This fibred product is k[x, Y, W, T]/(XY (1 + W)T), X = (x, x(1 + w)) Y = (y(1 + w), y)) W = (w, w) T = (xy, xy)

67 Section 4: Introducing singularities. 67 Idea. Attach these gluing automorphisms to straight lines emanating from singularities, along the invariant direction:

68 Section 4: Introducing singularities. 68 Idea. Attach these gluing automorphisms to straight lines emanating from singularities, along the invariant direction:

69 Section 4: Introducing singularities. 69 Problem. We then have compatibility problems for gluing at the vertices, so we have to extend the lines and continue to glue affine pieces using this automorphism.

70 Section 4: Introducing singularities. 70 Next problem. When two of these lines collide, we need the composition of automorphisms around the collision point to be the identity, to ensure compatibility of gluing.

71 Section 4: Introducing singularities. 71 Solution. (This is the main idea of [Kontsevich and Soibelman, 2004]). Add new lines with automorphisms emanating from collision point.

72 Section 4: Introducing singularities. 72

73 Section 4: Introducing singularities. 73

74 Section 4: Introducing singularities. 74

75 Section 4: Introducing singularities. 75

76 Section 4: Introducing singularities. 76

77 Section 4: Introducing singularities. 77 Main Theorem. Given an integral affine manifold B with singularities, polyhedral decomposition P, and strictly convex multivalued piecewise linear function ϕ, and given some local conditions on the singularities of B (local rigidity) then there exists a degeneration of Calabi-Yau varieties X Speck[[t]] controlled by this data. The degeneration is uniquely determined by some initial data (the log structure on the singular fibre), and is described by a union of affine hyperplanes with attached automorphisms, containing all tropical trees.

78 Section 4: Introducing singularities. 78 Remarks. The existence of such a deformation implies the existence of a flat deformation of the central fibre in the analytic category, i.e. a flat deformation X D, where D is a disk. The general fibre will be a compactification of X( ˇB 0 ), where ( ˇB, ˇϕ) is the discrete Legendre transform of (B, ϕ). In particular ˇX(B 0 ) = X( ˇB 0 ) (as topological spaces).

79 Section 4: Introducing singularities. 79 Remarks. The existence of such a deformation implies the existence of a flat deformation of the central fibre in the analytic category, i.e. a flat deformation X D, where D is a disk. The general fibre will be a compactification of X( ˇB 0 ), where ( ˇB, ˇϕ) is the discrete Legendre transform of (B, ϕ). In particular ˇX(B 0 ) = X( ˇB 0 ) (as topological spaces). It is this Legendre transform which allows us to replace Fukaya and Kontsevich, Soibelman s trees of gradient flow lines with tropical curves. These are much easier to control, which allows us to push these methods to all dimensions.

80 Section 4: Introducing singularities. 80 Remarks. The existence of such a deformation implies the existence of a flat deformation of the central fibre in the analytic category, i.e. a flat deformation X D, where D is a disk. The general fibre will be a compactification of X( ˇB 0 ), where ( ˇB, ˇϕ) is the discrete Legendre transform of (B, ϕ). In particular ˇX(B 0 ) = X( ˇB 0 ) (as topological spaces). It is this Legendre transform which allows us to replace Fukaya and Kontsevich, Soibelman s trees of gradient flow lines with tropical curves. These are much easier to control, which allows us to push these methods to all dimensions. Finally, given that the deformation X Speck[[t]] is described in terms of tropical trees, tropical rational curves emerge naturally in the calculation of periods, thus making a direct connection between the B side of mirror symmetry and tropical geometry.